Prediction of notch sensitivity effects in fatigue and in...

Prediction of notch sensitivity effects in fatigue and in environmentallyassisted cracking

J. T. P. DE CASTRO1, R. V. LANDIM2, J. C. C. LEITE3 and M. A. MEGGIOLARO1

1Mechanical Engineering Department, Pontifical Catholic University of Rio de Janeiro (PUC-Rio), Rio de Janeiro, Brazil, 2Instituto Nacional deTecnologia, INT, Rio de Janeiro, Brazil, 3Petrobras, Macaé, Brazil

Received Date: 26 September 2013; Accepted Date: 8 January 2014; Published Online: 7 March 2014

ABSTRACT Semi-empirical notch sensitivity factors q have been used for a long time to quantifynotch effects in fatigue design. Recently, this old concept has been mechanicallymodelled using sound stress analysis techniques, which properly consider the notch tipstress gradient influence on the fatigue behaviour of mechanically short cracks. This me-chanical model properly calculates q values from the basic fatigue properties of the mate-rial, its fatigue limit and crack propagation threshold, considering all the characteristicsof the notch geometry and of the loading, without the need for any adjustable parameter.This model’s predictions have been validated by proper tests, and a criterion to accepttolerable short cracks has been proposed based on it. In this work, this criterion is ex-tended to model notch sensitivity effects in environmentally assisted cracking conditions.

Keywords environmental effects; non-propagating cracks; notch sensitivity; short cracks.

NOMENCLATURE a = crack sizea0 = short crack characteristic size at R = 0aR = short crack characteristic size at R≠ 0b = notch depthE = Young’s modulusgr = grain sizeKf = 1 + q(Kt� 1), actual value of the stress concentration factor under fatigue loads

Kmax, Kmin = maximum and minimum values of the stress intensity factorKt = stress concentration factor

KtSCC = 1 + qSCC(Kt� 1), actual value of stress concentration factor under stress cor-rosion cracking conditions

KISCC = resistance to crack propagation under stress corrosion cracking conditionspz = plastic zoneq = notch sensitivity factor in fatigue

qSCC = notch sensitivity factor under stress corrosion cracking conditionsR = load ratio, R =Kmin/Kmax = σmin/σmax

SL = fatigue limit amplitudeSSCC = resistance to crack initiation under stress corrosion cracking conditionsSU = ultimate strengthSY = yield strengthγ = Bazant’s exponent

ΔK = stress intensity rangeΔKth0 = ΔKth(R = 0), long crack fatigue crack growth threshold at R = 0ΔKthR = ΔKth(R), long crack fatigue crack growth threshold at R≠ 0

ΔKthR(a) = short crack fatigue crack growth threshold at R, a function of the crack size aΔSL0 = range of the fatigue limit at R = 0ΔSLR = ΔSL(R) = 2SLR, range of the fatigue limit at R≠ 0

Correspondence: J. T. P. de Castro. E-mail: [email protected]

© 2014 Wiley Publishing Ltd. Fatigue Fract Engng Mater Struct, 2015, 38, 161–179 161

SPECIAL ISSUE CONTRIBUTION doi: 10.1111/ffe.12156

ΔS′L Rð Þ = range of the fatigue limit at R of standard unnotched and polished specimensΔσ = stress rangeη = free surface effect on KI

ρ = notch tip radiusσ = normal stress

σmax, σmin = maximum and minimum values of the stressvac = subscript denoting vacuum conditionsσn = nominal stress

φ(a) = effect of the stress gradient near a notch root on KI

I NTRODUCT ION

Fatigue damage is associated to two driving forces, onerelated to cyclic and the other to static damage mecha-nisms. In this way, fatigue crack growth (FCG) rates onany given environment depend on ΔK and Kmax, thestress intensity factor (SIF) range and maximum, or onany other pair of parameters related to them. In fact, itis more usual to use ΔK and R =Kmin/Kmax to describeand model FCG problems. Even though R is not a crackdriving force, such definitions are convenient from anoperational point of view because they are easier to com-pare with familiar notch sensitivity concepts long used byengineers, which this paper aims to improve.

To propagate long cracks by fatigue under fixed{ΔK, Kmax} or {ΔK, R} loading conditions, the appliedSIF range ΔK must be higher than the FCG thresholdat the given R ratio, ΔKth(R) =ΔKthR. Cracks may be con-sidered short while their FCG thresholds are smallerthan the long crack FCG threshold, thus while suchcracks can grow under ΔK<ΔKthR. This behaviour isnatural, because otherwise the stress ranges Δσ requiredto propagate short cracks at a given R would be higherthan their fatigue limits ΔSL(R) =ΔSLR, the stress rangeneeded to initiate and propagate cracks in smooth speci-mens at that R ratio. Indeed, assuming as usual that at anygiven fixed R ratio, the FCG process is driven by the SIFrange ΔK ∝Δσ√(πa), if very short cracks with size a→ 0had the same ΔKthR threshold the long cracks have, theywould need Δσ→∞ to grow by fatigue, a meaninglessrequirement.1–3 Such statements assume that the stressesare induced only by external loads; but if the cracks startfrom notch tips or from smooth surfaces previouslysubjected to plastic strain gradients or to any othersource that can induce residual stress fields, these resi-dent stresses must be added to the externally appliedstresses as static loading components that affect R butnot ΔK.

Microstructurally short cracks, those small comparedwith the grain size gr, are much affected by microstructuralbarriers such as grain boundaries; hence, cannot be wellmodelled for structural design purposes using macroscopicstress analysis techniques and isotropic properties.4–9

Mechanically short cracks, on the other hand, with sizesa> gr, may be modelled by Linear Elastic FractureMechanics (LEFM) concepts if the stress field that sur-rounds them is predominantly Linear Elastic, and if thematerial can be treated as isotropic and homogeneous insuch a scale. Because near-threshold FCG is always asso-ciated with small-scale yielding conditions, to check ifshort cracks really may be modelled in such a way, theidea is to follow Irwin’s steps by first assuming that suchconcepts are valid and then verifying if their predictionsare validated by proper tests. Hence, in the sequence,first LEFM techniques are used to develop a model forthe FCG behaviour of mechanically short cracks, inparticular those that depart from notches, and then thenotch sensitivity predictions based on it are corroboratedby proper experiments. Finally, such concepts are extendedto model notch sensitivity effects under environmentallyassisted cracking (EAC) conditions.

THE BEHAV IOUR OF SHORT CRACKS IN FAT IGUE

To reconcile the traditional fatigue (crack initiation)limit, ΔSL0 = 2SL(R = 0), with the FCG threshold of longcracks under pulsating loads, ΔKth0 =ΔKth(R = 0), Topperand his colleagues10–12 added to the physical crack size ahypothetical short crack characteristic size a0, a wise strata-gem that forces the SIF of all cracks, short or long, toobey the correct FCG limits:

ΔKI ¼ Δσffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiπ aþ a0ð Þ

p; where

a0 ¼ 1=πð Þ ΔKth0=ΔSL0ð Þ2:(1)

In this way, long cracks with a≫ a0 (in Griffith’splates under pulsating loads) do not grow by fatigue ifΔKI =Δσ√(πa)<ΔKth0, whereas very small cracks witha→0 do not grow if Δσ<ΔSL0, because ΔKI=Δσ√(πa0)<SL0√(πa0) =ΔKth0 in this case. Moreover, this clever ideareproduces the whole tendency of typical Δσj × aj datapoints in Kitagawa–Takahashi diagrams, where Δσj isthe stress range needed to propagate a fatigue crack withsize aj, see Fig. 1.13 This figure also shows the fatigue

162 J. T. P. DE CASTRO et al.

© 2014 Wiley Publishing Ltd. Fatigue Fract Engng Mater Struct, 2015, 38, 161–179

limit ΔSL0 and the stress range Δσ(a) =ΔKth0/√(πa)associated to the long crack threshold, which limit theregion that may contain non-propagating cracks, as wellas the El Haddad–Topper–Smith (ETS) curve, whichpredicts that cracks of any size should stop when

Δσ að Þ≤ ΔKth0=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiπ aþ a0ð Þ

p: (2)

Steels typically have 6<ΔKth0< 12MPa√m, ultimatetensile strengths 400< SU< 2000MPa, and fatigue limits200< SL< 1000MPa (the best high-strength steels withvery clean microstructures tend to maintain the trendSL≅SU/2 for smooth test specimens). Consequently,the range of their fatigue limits under pulsating loads(those with R = 0) estimated by Goodman is

ΔSL0 ≅ 2SUSL= SU þ SLð Þ⇒260 < ΔSL0 < 1300 MPa:

(3)

Hence, the range of characteristic short crack sizes insteel components (in large plates with a central crack sub-ject to pulsating tensile loads) estimated according to theETS model is

1=πð Þ� ΔKth0min=ΔSL0maxð Þ2≅7 < a0< 700μm≅ 1=πð Þ� ΔKth0max=ΔSL0minð Þ2: (4)

Because such a0 values are small, the denomination‘short crack characteristic size’ is justifiable. Indeed, theyhardly reach the detection thresholds of traditional non-destructive inspectionmethods 14. For typical Al alloys (with30< SL< 230MPa, 70< SU< 600MPa, 40<ΔSL0< 330MPa and 1.2<ΔKth0< 5MPa√m), the range estimatedfor a0 is a little larger, 1μm< a0< 5mm. So, it can beexpected that short crack effects on materials with highΔKth0 and low ΔSL0 to be more pronounced in Al alloys

than in steels. Note, however, that such values assume athrough-thickness one-dimensional (1D) crack, one thatcan be completely described by just one size parameter.Most such small cracks probably should be better treatedas two-dimensional (2D) cracks as discussed latter on; butto explain such concepts, this unidimensional analysis iscertainly more appropriate. Moreover, as the generic SIFof cracked structural components is given by KI=σ√(πa)�g(a/w), Yu, Duquesnay and Topper12 used the geometryfactor g(a/w) to generalize Eq. (1) and redefined the shortcrack characteristic size by

ΔKI ¼ Δσffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiπ aþ a0ð Þ

p�g a=wð Þ;where

a0 ¼ 1=πð Þ� ΔKth0= ΔSL0�g a=wð Þ� �� 2: (5)

The largest stress range Δσ that does not propagatemicrocracks in this case is also the fatigue limit, as itshould: if a≪ a0, ΔKI =ΔKth0⇒Δσ→ΔSL0. However,when the crack starts from a notch, as usual, its drivingforce is the stress range Δσ at the notch root, not thenominal stress range Δσn normally used in SIF expres-sions. As in such cases, the g(a/w) factor includes thestress concentration effect of the notch, it is better tosplit it into two parts: g(a/w) = η�φ(a), where φ(a)quantifies the effect of the stress gradient near the notchroot, which for microcracks tend towards Kt, that is,φ(a→ 0)→Kt, whereas the constant η quantifies the effectof the other parameters that affect KI, such as the freesurface. In this way, it is better to define a0 by

ΔKI ¼ η�φ að Þ�Δσnffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiπ aþ a0ð Þ

p;where

a0 ¼ 1=πð Þ� ΔK th0= η�ΔSL0ð Þ½ �2: (6)

The stress gradient effect quantified by φ(a) does notaffect a0 because the stress ranges at notch tips must besmaller than the fatigue limit to avoid crack initiation,Δσ(a→ 0) =KtΔσn = φ(0)Δσn<ΔSL0. However, becauseSIFs are crack driving forces, they should be materialindependent. Hence, the a0 effect on the short crackbehaviour should be used to modify the FCG thresholdsinstead of the SIF, making them a function of the cracksize, a trick that is quite convenient for operationalreasons. In this way, the a0-dependent FCG thresholdfor pulsating loads ΔKth(a, R = 0) =ΔKth0(a) becomes

ΔKth0 að ÞΔKth0

¼ Δσffiffiffiffiffiπa

p �g a=wð ÞΔσ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiπ aþ a0ð Þp �g a=wð Þ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffia

aþ a0

r⇒

ΔKth0 að Þ ¼ ΔKth0 1þ a0=að Þ½ ��1=2: (7)

Note that for a≫ a0 this short crack FCG thresholdtends to ΔKth0, the traditional long crack FCG threshold,and becomes independent of the crack size, as it should.

Fig. 1 Stress ranges Δσ(a) required to propagate cracks of size a un-der R = 0 in an HT80 steel plate with ΔKth0 = 11.2MPa√m andΔSL0 = 575MPa: long cracks, with a≫ a0, stop when Δσ≤ΔKth0/√(πa); whereas very short cracks, with a→ 0, stop when Δσ≤ΔSL0.The El Haddad–Topper–Smith model predicts that any crackshould stop when Δσ≤ΔKth0/√[π(a + a0)].

NOTCH SENS I T I V I TY 163


It may be convenient to assume that Eq. (7) is just one ofthe models that obey the long crack and the short cracklimit behaviours, introducing in the ΔK0(a) definition adata fitting parameter γ proposed by Bazant15 to obtain

ΔKth0 að Þ ¼ ΔKth0 1þ a0=að Þγ=2h i�1=γ

: (8)

Equation (8) reproduces the original ETS modelwhen γ = 2, as well as the bilinear limits shown in Fig. 1,Δσ =ΔSL0 and Δσ =ΔKth0/√(πa), when γ→∞. This addi-tional parameter may allow a better fitting of experimen-tal data, such as those collected by Tanaka et al.16 and byLivieri and Tovo,17 as shown in Fig. 2: most data on shortcracks are contained by the curves generated using γ = 1.5and γ = 8. The curves shown in Fig. 3 illustrate the influ-ence of γ on the minimum stress ranges needed to prop-agate short or long cracks under pulsating loads as afunction of the crack size a:

Δσ0 að Þ ¼ ΔKth0=ffiffiffiffiffiπa

p� �� 1þ a=a0ð Þγ=2h i�1=γ

: (9)

However, because fatigue damage depends on twodriving forces, ΔK and Kmax, as explained in the introduc-tion, Eq. (8) should be extended to consider the σmax

influence (indirectly modelled by the R ratio) on theshort crack behaviour. Thus, if ΔKthR =ΔKthR(a≫ aR, R)is the FCG threshold for long cracks, ΔSLR =ΔSL(R) isthe fatigue limit at the desired R ratio, and aR is the char-acteristic short crack size at R, then

ΔKthR að Þ ¼ ΔKthR� 1þ aR=að Þγ=2h i�1=γ

;whereaR ¼ 1=πð Þ ΔKthR= η�ΔSLRð Þ½ �2: (10)

Albeit defect-free microfilaments (whiskers) can bemade in lab conditions, structural components used inpractice always contain tiny defects such as inclusions,voids and scratches, which behave as small cracks. If thesize of such defects is not much smaller than a0, the struc-tural effects of such (mechanically) short cracks can beevaluated using LEFM concepts, as follows.17–27

I NFLUENCE OF SHORT CRACKS ON THE FAT IGUEL IM IT OF STRUCTURAL COMPONENTS

Traditional SN and εN methods are used to analyse anddesign supposedly crack-free components. However, asit is impossible to guarantee that they are really free ofcracks smaller than the detection threshold of the non-destructive method used to inspect them, SN or εN pre-dictions may become unreliable when such tiny defectsare introduced by any means during their manufactureor service. Hence, structural components should bedesigned to tolerate undetectable short cracks.

Despite self-evident, this prudent requirement is stillnot included in most fatigue design routines, which justintend to maintain the service stresses at critical pointsbelow their fatigue limits, Δσ<ΔSLR/φF, where φF is asuitable safety factor. Nevertheless, most long-life de-signs work just fine, thus they are somehow tolerant toundetectable or to functionally admissible short cracks.But the question ‘how much tolerant’ cannot be answeredby SN or εN procedures alone. Such problems can beavoided by adding a tolerance to short crack requirementto their ‘infinite’ life design criteria, which, in its simplestversion, may be given by

Δσ≤ΔKthR= φF �ffiffiffiffiffiπa

p �g a=wð Þ� 1þ aR=að Þγ=2h i1=γ� �

;

aR ¼ 1=πð Þ ΔKthR= ηΔSLRð Þ½ �2:(11)Fig. 2 The additional parameter γ in ΔKth0(a)/ΔKth0 = [1 + (a0/a)

γ/2]�1/γ

may allow a better fitting of the short crack FCG thresholds measuredexperimentally.

Fig. 3 Influence of γ in the fatigue limit curves Δσ0(a) predicted byEq. (9): the larger the γ value is, the faster Δσ0(a) tends to the bilin-ear limit defined by Δσ0 =ΔKth0/√(πa), the fatigue crack growththreshold under pulsating loads for long cracks with size a≫ a0,and to Δσ0 =ΔSL0, the fatigue limit under pulsating stresses for verysmall cracks with a≪ a0.



Because the fatigue limit ΔSLR already reflects theeffect of microstructural defects that are inherent to thematerial, Eq. (11) complements it by describing the toler-ance to cracks of size a (small or not) that may passunnoticed in actual service conditions. Such estimatescan be very useful for designers and quality control engi-neers. They can be used as a quantitative tool to evaluatethe effect of accidental damages to the surface of other-wise well-behaved components, but they have somelimitations. They assume that the short crack growsunidimensionally (1D), thus can be characterized by itssize a only. However, as the short cracks frequently aresmall compared with the structural component dimen-sions, they are better described as 2D cracks that growby fatigue in two directions maintaining their originalplane under mode I loads, but usually changing theirshape at every load cycle. Moreover, such estimates arevalid for mechanically but not for microstructurally shortcracks, that is, they are valid for cracks with both a and a0larger than the grain size gr. The local FCG behaviour ofmicrocracks with size a< gr is sensitive to microstruc-tural features such as the grain orientation and cannotbe properly modelled using macroscopic materialproperties. Such problems have academic interest,4–9

but as the grains still cannot be mapped in practice, they

cannot be properly used for structural engineering appli-cations yet.

To model short 2D (mechanical) cracks that tend togrow both in depth and width in the simplest possibleway, it is assumed that: (i) the cracks are loaded in puremode I under quasi-constant Δσ and R conditions, withno overloads or any other event capable of inducing loadsequence effects; (ii) material properties measured testing1D cracks in standard specimens such as ΔKthR may beused to simulate FCG or stress corrosion cracking

(SCC) behaviour of 2D cracks; and (iii) 2D surface orcorner cracks can be well modelled as having an approx-imately elliptical front, thus their SIF can be described bythe classical Newman–Raju equations.26,28–30 If such rea-sonable hypotheses hold as expected, then the structuralcomponents tolerance to short or long fatigue cracksare given by

Δσ <

ΔKthR=ffiffiffiffiffiπa

p �Φa a; c;w; tð Þ� 1þ aR=að Þγ=2h i1=γ� �

ΔKthR=ffiffiffiffiffiπc

p �Φc a; c;w; tð Þ� 1þ aR=cð Þγ=2h i1=γ� �

8>>><>>>:

(12)

The mode I SIFs at the tips of the semi-axes a along thedepth and c along the width of semi-elliptical surface cracksin a plate of width 2w and thickness t loaded under a puretensile nominal load σ are KI(a) =σ√(πa)�Φa =σ√(πa)�F�M/Q0.5 and KI(c) =σ√(πa)�Φc =σ√(πa)�(F�M/Q0.5)�(a/c)�G for a< t and c<w (Eq. (13)). The similar mode I SIFsfor quarter-elliptical corner cracks are even morecomplex.28 Such complex 2D SIFs enhance the operationaladvantage of treating the FCG threshold as a function ofthe crack size, ΔKthR(a).

However, if the short 2D cracks start from notch tips,as usual, the stress analysis problem may be still morecomplex. In general, they must include non-negligible3D gradient effects around the notch tips, as discussedelsewhere.29 On the other side, the tolerance to SCCcracks can be treated using these same principles, byproperly changing the fatigue properties ΔKthR and ΔSLRby the corresponding material resistances to SCC crack-ing in the desired environment, KISCC and SSCC, asexplained later on.

KI að Þ ¼ σffiffiffiffiffiπa

p �F�M=ffiffiffiffiQ

p

KI cð Þ ¼ σffiffiffiffiffiπc

p �F� M=ffiffiffiffiQ

p� ��a=c�GF c=w; a=tð Þ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisec πc=2wð Þ ffiffiffiffiffiffiffi

a=tph ir

� 1� 0:025 c=wð Þ ffiffiffiffiffiffiffia=t

ph i2þ 0:06 c=wð Þ ffiffiffiffiffiffiffi

a=tph i4

M ¼1:13� 0:09

acþ 0:89

0:2þ a=c� 0:54

a2

t2þ 0:5� 1

0:65þ a=cþ 14 1� a

c

� �24

a4

t4; a≤c

c=aþ 0:04 c=að Þ2 þ c=að Þ4:5 a=tð Þ2 0:2� 0:11 a=tð Þ2h i

; a > c

8>><>>:

Q ¼1þ 1:464 a=cð Þ1:65; a≤c

1þ 1:464 c=að Þ1:65; a > c; G ¼

1:1þ 0:35 a=tð Þ2; a≤c

1:1þ 0:35 a=tð Þ2 c=að Þ; a > c

8<:

8<:

8>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>:

(13)



ANALYS IS OF NOTCH SENS I T IV I TY EFFECTS ONFAT IGUE

The SIF of small mechanical cracks that start at the rootsof notches with depth b and tip radius ρ can be estimatedby KI≅ 1.12σn�f1(Kt, a)√(πa), where f1 = σy(x)/σn is thestress concentration perpendicular to the crack plane atthe point (x = b + a, y = 0) ahead of such tips. The stressconcentration effect of such notches can be estimatedby an ellipse with semi-axes b and c and notch tip radiusρ = c2/b. If the ellipse axis 2b is centred at the x-axis originand is perpendicular to the nominal stress σn, then31

f 1 ¼σ y x ¼ bþ a; y ¼ 0ð Þ

σn¼

1þb2 � 2bc� �

x�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 � b2 þ c2

p� �x2 � b2 þ c2� �þ bc2 b� cð Þx

b� cð Þ2 x2 � b2 þ c2� � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

x2 � b2 þ c2p :

(14)

The high stress gradient ahead of elongated notch tipsjustifies the peculiar behaviour of short cracks that startfrom sharp notches: in the Linear Elastic case, thetangential stress at x = 1.2b ahead any elliptical hole withb≥ c is σy(1.2, 0)/σn≅ 2, independently of the ellipticalnotch stress concentration factor (SCF) Kt. As the stressgradient around sharp notch tips is high, the SIF inducedby remotely applied loads on short cracks that start therefirst grows fast with their growing sizes a, but after asmall Δa increment they may stabilize or even decreasetheir rates for a while before growing once again, sincethe notch effect on KI may diminish sharply as the shortcrack grows. Indeed, the term √a that increases KI = 1.12σ√(πa)�f1 can be overcompensated by the abrupt fall in f1near the notch tip (Fig. 4). Such simple concepts can beused to evaluate the tolerance to fatigue cracks that startfrom such notches by using the KI(a) and ΔKthR(a) esti-mates for the crack SIF and for the FCG threshold ofthe material26. In other words, short cracks can bearrested whenever their SIFs, which are highly sensitive

to the stress gradient ahead of the notch tips, becomesmaller than the short crack FCG threshold at the givenR ratio, which depends on the crack size a, whereas it isnot much larger than the characteristic short crack sizeaR: ΔKI(a)≤ΔKthR(a)⇒ crack arrest.

The notch sensitivity q still is quantified for designpurposes by empirical curves fitted to only seven experi-mental points compiled by Peterson32 a long time ago.It is used to estimate fatigue limits measured underfixed Δσn and R in notched test specimens with a SCFKt≥Kf = 1 + q�(Kt� 1) by ΔSLR ¼ ΔS′

L Rð Þ=Kf , where ΔSLRis the fatigue limit of the notched test specimens, andΔS′

L Rð Þ is the unnotched fatigue limit under the same Rratio.

However, according to Frost,33 early data showingthat small non-propagating fatigue cracks are found atnotch tips when ΔSLR/Kt<Δσn<ΔSLR/Kf goes back asfar as 1949. It is thus reasonable to expect that q is relatedto the fatigue behaviour of short cracks emanating fromnotch tips or that such tiny cracks can be used to quantifyKf≤Kt values. The notch sensitivity can in fact be calcu-lated in this way using relatively simple but soundmechanical principles that do not require heuristic argu-ments, neither arbitrary fitting parameters. To start with,according to Tada,34 the SIF of a crack with size a thatdeparts from a circular hole of radius ρ is given within1% by

Note that when a→ 0⇒ x→ 0, this equation tends tothe expected limit,

lima→0

ΔKI ¼ 1:1215�3�Δσ ffiffiffiffiffiπa

p: (16)

Fig. 4 KI≅ 1.12�σn√(πa)�f1(Kt, a) estimate for small cracks a≤ b/5that start from the tips of several Inglis holes with b = 10mm.

KI ¼ 1:1215�σ ffiffiffiffiffiπa

p �φ xð Þ; x≡a=ρ

φ xð Þ ¼ 1þ 0:21þ xð Þ þ

0:31þ xð Þ6

" #� 2� 2:354� x

1þ x

�þ 1:206� x

1þ x

�2

� 0:221� x1þ x

�3" #8><

>: (15)



Indeed, this equation combines the solution for anedge crack in a semi-infinite plate with the SCF of theKirsch hole, Kt = 3. Note also that it reproduces the cor-rect limit once again when a→∞, the SIF of an Irwin’splate with a crack of length a (in fact, a + 2ρ≅ a, as in thiscase a≫ ρ) because the crack tip is so distant from thehole that it is not affected by it:

lima→∞

ΔKI ¼ Δσffiffiffiffiffiffiffiffiffiffiπa=2

p: (17)

Hence, for Kirsch holes such limits are φ(x = 0) = 3 andφ(x→∞) = 1/1.1215√2≅ 0.63. The FCG condition forcracks that start at such notch borders under pulsatingloads is thus

ΔKI ¼ Δσffiffiffiffiffiπa

p �η�φ a=ρð Þ > ΔKth0 að Þ¼ ΔKth0� 1þ a0=að Þγ=2

h i�1=γ(18)

where ΔKth0 =ΔSL0√(πa0)≡ΔKth0(a≫ a0), and a0 = (1/π)[ΔKth0/(η�ΔSL0)]2. Note that a0 cannot depend on thestress gradient factor φ(a/w), otherwise it would not bea material property. This FCG criterion can be rewrittenusing two dimensionless functions, one related to thenotch stress gradient φ(a/ρ), and the other g(ΔSL0/Δσ,a/ρ, ΔKth0/ΔSL0√ρ, γ), which includes the effects of theapplied stress range Δσ, the crack size a, the notch tipradius ρ, the fatigue resistances ΔSL0 and ΔKth0, andthe optional data fitting exponent γ (if it is used):25

φ a=ρð Þ > ΔSL0=Δσð Þ� ΔKth0= ΔSL0ffiffiffiρ

p� �� η

ffiffiffiffiffiffiffiffiffiffiπa=ρ

p� �γþ ΔKth0= ΔSL0

ffiffiffiρ

p� �� γh i1=γ≡g

ΔSL0

Δσ;aρ;ΔKth0

ΔSL0ffiffiffiρ

p ; γ �

: (19)

Therefore, if x≡ a/ρ and κ≡ΔKth0/(ΔSL0√ρ) =η�√(πa0/ρ), a fatigue crack departing from a Kirsch holeunder pulsating loads grows whenever φ(x)/g(ΔSL0/Δσ, x,κ, γ)> 1. Figure 5 plots some φ/g functions for severalfatigue strength to loading stress range ratios ΔSL0/Δσas a function of the normalized crack length x for a smallnotch radius ρ≅ 1.40a0, comparable with the short crackcharacteristic size, and for κ =ΔKth0/[ΔSL0√(1.4a0)] =1.12√(π/1.4) = 1.68 and γ = 6.26

For high applied stress ranges Δσ, the strength to loadratio ΔSL0/Δσ is small, and the corresponding φ/g curve isalways higher than 1, so cracks will initiate and propagatefrom this small Kirsch hole border without stopping dur-ing this process. One example of such a case is the uppercurve in Fig. 5, which shows the function φ/g1.4 obtainedfor ΔSL0/Δσ = 1.4. On the other hand, small stress rangeswith load ratios ΔSL0/Δσ≥Kt = 3 have φ/g< 1, meaning

that such loads cannot initiate a fatigue crack from thishole and that small enough cracks introduced there byany other means will not propagate at such low loads.This is illustrated by curves φ/g3, associated with the limitcase where the local stress range equals the materialfatigue strength ΔSL0/Δσ = 3, and φ/g4, associated with astill smaller load, ΔSL0/Δσ = 4.

Three other curves must be analysed in Fig. 5. Theφ/g2.3 curve crosses the φ/g = 1 line once, meaning thatsuch an intermediate load can initiate and propagate afatigue crack from this hole border, until the decreasingφ/g2.3 ratio reaches 1, where the crack stops because thestress gradient ahead of its small tip is sharp enoughto eventually force ΔKI(a)<ΔKth(a). Thus, under thisΔσ =ΔSL0/2.3 loading, a non-propagating fatigue crackis generated at this small hole border, with a size givenby the corresponding a/ρ abscissa where φ/g2.3 = 1. Theφ/g1.85 curve intersects the φ/g = 1 line twice. This loadlevel also generates a fatigue crack at the hole border,which will propagate until reaching the maximum sizeobtained from the abscissa of the first intersection point(on the left in that figure), where the crack stops becauseit reaches ΔKI(a)<ΔKth0(a). Moreover, cracks longerthan the second intersection point will restart propagat-ing by fatigue under Δσ =ΔSL0/1.85, until eventuallyfracturing this Kirsch plate. However, the crack initiatedby fatigue under such an intermediate pulsating loadrange cannot propagate between these two intersectionpoints by fatigue alone, if the loading parameters {Δσ,σmax} remain constant. Hence, the crack can only growin this region if helped by a different damage mecha-nism, such as SCC or creep.

The FCG behaviour of these two curves seems differ-ent in Fig. 5, yet they are similar. Indeed, the φ/g2.3curve crosses the φ/g = 1 line twice if the graph isextended to include larger cracks.26 This is so because asufficiently long crack can always propagate by fatigueunder any given (even if small) Δσ range if its SIF range

Fig. 5 Cracks that can start from the border of a (small) Kirsch holewith Kt = 3 may propagate by fatigue and then stop if their φ/g< 1(ρ≅ 1.40a0, κ = 1.68, and γ = 6 in this figure).



ΔK =Δσ√(πa)�g(a/w) grows with the crack size a, as inthis Kirsch plate. In fact, all φ/g curves eventually becomehigher enough for sufficiently large a/ρ values, even thosethat cannot initiate a crack by fatigue, such as φ/g4.

Finally, note the φ/g1.64 curve that is tangent to theφ/ = 1 line in Fig. 5. Therefore, this pulsating stressrange Δσ =ΔSL0/1.64 is the smallest one that can causecrack initiation and growth (without arrest) from thatnotch border by fatigue alone. Hence, by definition,the fatigue SCF of this small Kirsch hole (withρ≅ 1.40�a0, κ = η�√(πa0/ρ) = 1.5 and γ = 6) is thus Kf = 1.64;therefore, its notch sensitivity factor is q = (Kf� 1)/(Kt� 1)= (1.64� 1)/(3� 1) = 0.32. Moreover, the abscissa of thetangency point between the φ/g1.64 curve and the φ/g = 1line gives the largest non-propagating crack size that canarise from it by fatigue alone, amax. For any other ρ/a0, γand κ = η�√(πa0/ρ) combination, Kf and amax can alwaysbe found by solving the system

φ=g ¼ 1∂ φ=g� �

=∂x ¼ 0⇒

(φ xmaxð Þ ¼ g xmax;Kf ; κ; γ

� �∂φ xmaxð Þ=∂x ¼ ∂g xmax;Kf ; κ; γ

� �=∂x

(

(20)

Kirsch (circular) holes cause relatively mild stress gradi-ents. Larger holes compared with the short crack charac-teristic size, ρ≫ a0, are associated to small κ = η�√(πa0/ρ)values and do not induce short crack arrest. For example,Kirsch holes with ρ> 7�a0 in a material with γ = 6 do notinduce non-propagating fatigue cracks under fixedpulsating loads, thus have q = 1. That is a sound mechan-ical interpretation for the notch sensitivity concept. If fora given γ Eq. (20) is solved for several notch tip radii ρusing κ≡ΔKth0/ΔSL0√ρ, then the notch sensitivity fac-tor q is obtained by

q κ; γð Þ≡ Kf κ; γð Þ � 1� �

= Kt � 1ð Þ: (21)

This approach has four major assets: (i) it is an analyt-ical procedure; (ii) it considers the effect of the fatigue re-sistances to crack initiation and propagation on q; (iii) itcan use the exponent γ to generalize the original ETSmodel, but it does not need it neither any other data

fitting parameter; and (iv) it can be easily extended to dealwith other notch geometries. For example, the SIF ofcracks that depart from a semi-elliptical notch withsemi-axes b and c, with b in the same direction of the cracka, which is perpendicular to the (nominal) stress Δσ, canbe described by:

ΔKI ¼ η�F a=b; c=bð Þ�Δσ ffiffiffiffiffiπa

p(22)

where η = 1.1215 is the free surface correction factor andF(a/b, c/b) is the geometrical factor associated to thenotch stress concentration effect. Such notch SCFs Kt

are approximately given by:34

Kt ¼ 1þ 2b=cð Þ� 1þ 0:1215= 1þ c=bð Þ2:5h i

: (23)

Using s = a/(a + b), two analytical expressions for F(a/b,c/b) were introduced in Ref. [25] by fitting resultsobtained by a series of Finite Element (FE) analyses forseveral types of semi-elliptical notches, made using theQUEBRA2D software,35 which reproduce very well theresults of Nishitani and Tada quoted by Bazant15 (Fig. 6):

Equating g =φ, the minimum stress range needed tostart and propagate a fatigue crack from the edge of suchnotches can be calculated for several combinations of κand γ, leading to expressions for Kf and, consequently,

Fig. 6 Stress intensity factors calculated for cracks that depart fromsemi-elliptical notches with c≤ b.

F a=b; c=bð Þ≡f K t; sð Þ ¼ Kt

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� exp �K2

t �s� ��

= K2t �s

� �q; c≤b

F a=b; c=bð Þ≡f ′ Kt; sð Þ ¼ Kt

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� exp �K2

t �s� �

K2t �s

s� 1� exp �K2

t

� �� s=2; c≥b

8>>><>>>:

(24)



for q (Fig. 7). Note that the notch sensitivity q(1/κ) esti-mated in this way is almost linear for sensitivities q> 0;hence, it can be approximately modelled by

q κ; γð Þ≅q1 γð Þ=κ � q0 γð Þ ¼ q1 γð ÞΔSL0ffiffiffiρ

p=ΔKth0 � q0 γð Þ:

(25)

The parameters q0(γ) and q1(γ) that fit the quasi-linearpart of q(γ, κ) depend only on γ, whereas the parameter1/κ =ΔSL0√ρ/ΔKth0 includes the material fatigue limitand FCG threshold, as well as the notch tip radius ρ.Note that Eq. (25) predicts q> 1 for high 1/κ values, ifthe notch has a large tip radius ρ compared with the a0value, larger than a radius ρsup given by

ΔSL0ffiffiffiffiffiffiffiffiρ sup

pΔKth0

>1þ q0 γð Þq1 γð Þ ⇒ρ sup >

1þ q0 γð Þq1 γð Þ �ΔKth0

ΔSL0

�2

:

(26)

Equation (26) is written for pulsating loads (R = 0), butit can be easily generalized for any other R ratio. It mayseem strange to predict a notch sensitivity q> 1 (as for fa-tigue design and similar applications q = 1 must be used insuch cases because by definition Kf≤Kt), but such valueshave a good physical interpretation: sensitivities q> 1mean that the cracks initiated by fatigue under fixed loadconditions {Δσ, σmax} from the notch border do not stop,thus never become non-propagating under such condi-tions. This occurs when the stress gradient near thenotch tip is too gentle to affect the short crack behaviour.In fact, in the absence of compressive residual stresses,the only mechanical reason for cracks induced by fatiguefrom a notch border to stop after growing for a while(under the same fixed load conditions that initiated them)

inside an isotropic material is the stress gradient near thenotch tip. To stop a crack, the stress range decreaseahead of the notch tip induced by the stress gradientaround it must be able to surpass the SIF increase in-duced by the crack size increment, in such a way thatΔK = η�φ(a)�Δσ√(πa) can decrease as a grows until be-coming smaller than the propagation threshold ΔKth0(a)(or ΔKthR(a) for R≠ 0). The crack stop size ast (underpulsating loads) is thus reached when

ΔKI ¼ η�φ astð Þ�Δσ ffiffiffiffiffiffiffiffiπast

p ¼ ΔKth0 að Þ¼ ΔKth0� 1þ a0=astð Þγ=2

h i�1=γ: (27)

Equation (25) can also possibly predict q< 0, that is,negative notch sensitivities (up to q≅�0.2 in the Kirschhole case, see Fig. 7), which seems even more strangethan the q> 1 values. This occurs when the notch is toosharp, with a tip radius ρinf given by

ΔSL0ffiffiffiffiffiffiffiffiρ inf

p=ΔKth0 < q0 γð Þ=q1 γð Þ⇒

ρ inf< q0 γð Þ� �=q1 γð Þ� ΔKth0=ΔSL0ð Þ� �2

:

(28)

However, values q< 0 also have a clear physical inter-pretation: in such cases, it is easier to start a fatigue crackfrom a notchless surface than from the notch border.This occurs because the stress gradient ∂g(a)/∂a nearthe notch tip is so large that the SIF of the crack quicklyreaches the long crack condition limit, which does not in-clude any more the free surface factor η = 1.12, which af-fects the SIF of the cracks that start from unnotchedsurfaces. In most materials, the value of ρinf is on the or-der of a few micrometres,36 meaning that small internaldefects with radii ρ< ρinf are not harmful, hence thatthe cracks will start, as usual, at the free surface of thepiece.

Traditional semi-empirical notch sensitivity estimates,such as Peterson’s q = (1 + α/ρ)�1, based on an ill-definedlength parameter α obtained by fitting only seven exper-imental points, suppose that the notch sensitivity dependsonly on the notch tip radius ρ and on the steel tensilestrength SU, and only on ρ for Al alloys. The model pro-posed here, on the other hand, recognizes that q dependson ρ, ΔSL0, ΔKth0, γ, and also on the component andnotch geometries, which affect the stress gradient aheadof the notch tip. There are reasonable relations betweenΔSL0 and SU, but none between ΔKth0 and SU. Thismeans that two steels of the same SU, but very differentΔKth0, should behave identically according to Peterson-like q estimates, whereas they usually do not. To quantifysuch evaluations, 450 steels and Al alloys with reportedSU, SL(R =�1) and ΔKth0 values were gathered in VIDA’sdatabase.25,26,37 The steels set was separated in 400, 800,

Fig. 7Notch sensitivity q(1/κ) estimated for a (circular) Kirsch hole.



1200, 1600 and 2000MPa strength ranges, to use theiraverage fatigue limits and FCG thresholds in the notchsensitivity analyses. All Al alloys were analysed with re-spect to their average strength SU = 225MPa. AssumingR =�1, their average fatigue limits were used to calculatea0. The values so obtained were used to produce q × ρcurves for the Kirsch hole (Fig. 8), assuming a typicalvalue γ = 6. Note how such curves reproduce quite wellthe curves proposed by Peterson a long time ago.32

Notch sensitivities for semi-elliptical notches in Alalloys, estimated using their average fatigue propertiesmentioned previously, are shown in Fig. 9. They dependon ρ, ΔSL(R =�1)≅ SL, ΔKth0, γ and also on their c/b ra-tios. Thus, they depend also on the notch shape, not only

on their tip radii ρ, as assumed in traditional SN analyses.As a matter of fact, they slightly depend on the R ratio aswell, because it affects a little bit the ΔKthR/SLR ratio.Because the c/b ratio effect is very significant, it cannot beignored in practice. The notch sensitivity of steels can becalculated in the same way, and it follows a similar pattern.Therefore, the observations made earlier for the parametersthat control q in Al alloys are valid for steels as well. Furtherdetails on such calculations can be found in Ref. [25].

A different approach to model the notch sensitivityproblem, called the theory of critical distances (TCD),is explored elsewhere.38–41 Predictions made by theTCD model are similar but not identical to the predic-tions obtained by the stress gradient model proposedhere. The TCD generalizes Peterson’s and Neuber’soriginal ideas, but the authors believe the stress gradientmodel is based on clearer mechanical bases and is easierto apply to notched components. The effect of micro-structural defects on fatigue strength is deeply studiedby Murakami.36

EXPER IMENTAL VER I F ICAT ION OF THE NOTCHSENS I T IV I TY PRED ICT IONS ON FAT IGUE

Stop holes are widely used as an emergency crack repairtechnique. The hole is drilled at the crack front to re-move its tip and to force it to re-initiate before restatingits growth process. This simple trick may increase thecracked component durability, but its efficiency dependson several variables, among them the stop-hole radius ρ.To quantify how beneficial such holes can be, 23precracked SE(T) test specimens with width w = 80mmand thickness t = 8mm (Fig. 10) were repaired in thisway and then fatigue tested under constant force ampli-tudes at R = 0.57.42 They were all cut from a plate of a6082T6 Al alloy (0.7–1.3Si, 0.6–1.2Mg, 0.4–1.0Mn,0.5Fe, 0.25Cr, 0.2Zn, 0.1Cu and 0.1Ti) with yieldstrength SY = 280MPa, SU = 327MPa, Young’s modulusE = 68GPa, reduction in area RA = 12% and Vickershardness HV50 = 95 kg/mm2, in its Longitudinal-Transversal (LT) direction. This alloy is used, for example,in vehicles, railway components and shipbuilding.

The fatigue tests were all performed at 30Hz on a100 kN computer controlled servo-hydraulic machine,following ASTM E647 procedures. The high R = 0.57was chosen to avoid crack closure effects. After pre-cracking a specimen, a stop hole with radius ρ = 1, 2.5,or 3mm was carefully centred and drilled at its cracktip. To do so, the cracked specimens were removed fromthe testing machine, fixed on a milling machine, drilledat low feedings with plenty refrigeration and thenfinally reamed to generate elongated notches with a tipdiameter accuracy of 1.5μm, all with the same initial size

Fig. 9 Notch sensitivity q as a function of the tip radius ρ of semi-elliptical notches inAl alloys, estimated using a0 = (1/π)(ΔKth0/1.12SL0)

2 =264μm, SU=225MPa and γ =6.

Fig. 8 Notch sensitivity q for Kirsch holes, estimated using meanΔKth0 and ΔSL0 values from 450 steels and Al alloys, supposingγ = 6. Note that q = 0 means that it is easier to initiate a crack froma free surface than from the border of very small holes.



ai = 27.5mm⇒ ai/w = 0.344, to avoid interference of possi-ble crack length or residual ligament rl =w� ai effects.Great care was taken to avoid introducing residualstresses around the stop hole by any means during itsdrilling and reaming process. Finally, the test specimenswere remounted on the test machine, and the fatigue testwas restarted under the previous loading conditions.Figure 11 also shows typical data obtained from suchrepaired specimens. Table 1 summarizes the testing con-ditions after introducing the stop holes, and the fatiguelife increments obtained from them. The pseudo SIFsof the repaired specimens listed in this table, ΔK* = 1.895ΔP/t√w, were calculated using the SIF expression for the

SE(T) with identical a/w, given by43

The fatigue crack re-initiation lives at the stop-hole tipscan be reliably modelled by traditional εN procedures. Thismodelling process requires the cyclic properties of the6082T6 Al alloy, that is, Ramberg–Osgood’s k′ = 443MPaand n′=0.064 and Coffin–Manson’s σf′=485MPa,b =�0.0695, εf′ = 0.733 and c =�0.827;44 the nominal stresshistory (Table 1); and the SCF of the notches generated bythe stop holes. These can be estimated by Creager andParis,45 giving, for example, Kt≅ 12.38 for a stop hole withradius ρ = 1, or else by Inglis,46 Kt≅ 1 + 2√(a/ρ) = 11.49 inthis case; but the resulting notch SCFs were instead calcu-lated by finite elements: Kt = 11.8, 8.1 and 7.6 for ρ = 2.5

and 3mm, respectively. The life improvement induced bythe stop holes can be estimated by calculating stresses andstrains at their borders byNeuber’s rule, and then the crackre-initiation lives considering mean load effects. Sucheffects cannot be neglected, because the R ratio used inthe tests was high. In fact, Coffin-Manson predictions arehighly non-conservative, thus useless in this case.Figures 11–14 show that the lives predicted by MorrowEL and by Smith–Watson–Topper (SWT) are similar inthis case (but such a similarity cannot be assumed before-hand; because in many other cases, these rules can predictvery different fatigue lives).26

The lives predicted using Kt for the two larger stop-holes reproduced reasonably well the tests results, seeFig. 11 for the ρ = 3mm results. However, the lives pre-dicted using Kt for the ρ = 1mm stop hole shown inFig. 12 are too conservative in comparison with the mea-sured data. Such better-than-predicted fatigue lives ofcourse do not mean that the smaller hole is more efficient

Fig. 10 Test specimen used to test the stop-hole size effect on their efficiency as a crack repair method and typical results obtained with them.

KI ¼ Pt

ffiffiffiffiw

p 1:99aw

h i0:5� 0:41

aw

h i1:5þ 18:7

aw

h i2:5� 38:85

aw

h i3:5þ 53:85

aw

h i4:5 �: (29)

Fig. 11 Fatigue crack re-initiation lives measured for the largerstop-holes with tip radius ρ = 3.0mm and fatigue lives predicted byεN procedures using the resulting elongated notch stress concentra-tion factor Kt calculated by FE in Neuber’s rule (ΔK* is the pseudo-stress intensity factor applied on the notch).



than the larger ones, as the larger stop-holes are associ-ated with longer fatigue crack re-initiation lives for agiven load, see Table 1. Therefore, from a modelingpoint of view, the main result obtained from such figuresis that εN life predictions made using traditionalprocedures based on Kt, Neuber, and Morrow orSmith–Watson–Topper were satisfactory for the larger

stop-holes, but severely underestimated the re-initiationlives for the smaller ones.

The better-than-expected fatigue lives obtained fromthe smaller stop-holes could be due to compressive resid-ual stresses. However, all stop holes were drilled andreamed following identical procedures, and their diame-ters were all large enough to remove the previous cracktip plastic zones, leaving only virgin material ahead oftheir tips. Hence, because the larger stop-hole lives werewell predicted, supposing σres = 0, it is difficult to justifywhy high compressive residual stresses would be presentonly around the smaller stop-hole tips. The same canbe said about the stop holes’ surface finish. However,the smaller stop-holes generate larger SCF than thelarger ones; therefore, they induce a steeper stress gradi-ent ahead of their tips. This effect can significantly affectthe growth of short cracks and, consequently, the fatiguenotch sensitivity of the stop hole. Indeed, when using theproperly calculated fatigue SCF Kf instead of Kt with thetraditional εN procedures, considering the elongatednotch sensitivity q by the method proposed here, allestimated fatigue crack re-initiation lives reproduce quitewell the measured results (Figs 13 and 14). The Al6082T6 fatigue limit and fatigue crack propagationthreshold under pulsating loads (R = 0) needed tocalculate Kf are estimated as ΔKth0 = 4.8MPa√m and

Table 1 Crack re-initiation lives Nr after introducing the stop hole at their tips

ρ = 1mm ρ = 2.5mm ρ = 3mm

ΔK*MPa√m Nr×103 cycles ΔK*MPa√m Nr×103 cycles ΔK*MPa√m Nr×103 cycles

6.0 >2000 7.5 >2000 8.5 >20007.4 980, 724, 580 8.1 1800 9.0 1150, 9608.0 600, 560, 510 10.1 355, 270 10.1 611, 58010.1 119, 84 13.5 65, 58, 37 14.0 60, 32

Fig. 12 Similar to Fig. 11, but for the smaller stop-holes with tipradius ρ = 1.0mm.

Fig. 13 Fatigue crack re-initiation lives measured for the stop-holeroot with radius ρ = 3.0mm and predicted using the resulting elon-gated notch fatigue stress concentration factor Kf calculated by theprocedures proposed in this paper in Neuber’s rule (ΔK* is thepseudo-stress intensity factor applied on the notch).

Fig. 14 Similar to Fig. 13, but for the smaller stop-holes with tipradius ρ = 1.0mm and Kf<Kt.



ΔSL0 = 110MPa, following traditional structural designpractices, and γ = 6.

Note that the Kt-based predictions shown in Fig. 11are very similar to the Kf -based predictions shown inFig. 13, because the larger stop-holes have q≅ 1. How-ever, the Kf -based life predictions for the smallerρ = 1mm stop-holes shown in Fig. 14 are much betterthan the Kt-based predictions shown in Fig. 12. Note alsothat the word prediction can in fact be used here, becausethe curves shown in such figures result from fatiguecrack re-initiation life estimates made using only me-chanical principles and material data obtained from theliterature, without considering any of the measured datapoints. Hence, they are really predicted, not data-fittedcurves. Moreover, an additional test made after thosecalculations confirmed the prediction that the ρ = 1mmstop-hole could tolerate a higher pseudo-SIF rangeΔK* = 7MPa√m, as indeed it did (Fig. 14).

NOTCH SENS I T IV I TY EFFECTS ON ENV IRON-MENTALLY ASS ISTED CRACK ING

Environmentally assisted cracking involves the nucleationand/or propagation of cracks in susceptible materialsimmerged in aggressive media. This time-dependentchemical/mechanical damage mechanism may eventuallylead to fracture under static tensile stresses that might bewell below the material strength in benign environments.EACmechanisms are usually subdivided into the followingsub-mechanisms: SCC, due to chemical reactions at cracktips enhanced by the high stresses that surround them;hydrogen embrittlement, due to high hydrostatic stressesinduced by penetration of small H atoms inside the gapsof crystalline lattices and/or grain boundaries; liquid metalembrittlement (LME), due to the interaction of liquidmetals such as Hg, Pb, Ga, Cd or Zn with tensionedsurfaces of LME-sensitive structural alloys; solid metalembrittlement in tensioned coatings or inclusions; and cor-rosion fatigue, due to a synergic interaction between cyclicloads and electrochemical reactions at crack tips. SuchEAC mechanisms may have different phenomenologies,but they all have a common feature: unlike other corrosionproblems, they depend both on the environment/materialpair and on the stress state, because cracks cannot grow un-less loaded by tensile stresses. Hence, all EACmechanismsrequire an EAC-sensitive material, an aggressive mediumand tensile stresses.47–58

Indeed, cracks only grow if driven by tensile stresses, andthe environment contribution is to decrease the material re-sistance to the cracking process. As the terminology stresscorrosion cracking enhances this mutual dependence, it is pre-ferred here to name all EAC mechanisms as SCC whenthere is no need separate them. Such problems are

important for many industries, because costs and deliverytimes for special SCC-resistant alloys are large and keep in-creasing. Major SCC problems occur, for example, in theoil industry, because oil and gas fields can contain consider-able amounts of H2S, which may attack steel pipelines, andin the aeronautical industry, when their light aluminumstructures must operate in saline environments, such as inaircraft carriers, offshore platforms or coastal airports.

However, for structural analysis purposes, most SCCproblems have been treated so far by a simplisticoverconservative policy on susceptible material-environmentpairs: if aggressivemedia are unavoidable during the servicelives of structural components, the standard design solutionis to choose a material resistant to SCC in such media tobuild them. A less expensive alternative solution may beto recover the structural component surface with a suitablenobler coating, if such a coating is available. SCC-proofcoatings must be properly adherent, scratch resistant andmore reliable than common corrosion-resistant coatings,because structural components can fail without warning un-der such conditions. Such overconservative design criteriamay be safe, but they can also be too expensive if an otherwiseattractive material is summarily disqualified in the designstage when it may suffer SCC in the service environment,without considering any stress analysis issues. Nevertheless,the SCC behaviour cannot be properly evaluated neglectingthe influence of the stress fields that drive them. Decisionsbased on such an inflexible pass/fail approach may causesevere cost penalties, because no crack can growunless drivenby a tensile stress caused by the service loads and by the resid-ual stresses induced by previous loads and overloads.

In other words, although EAC conditions may be diffi-cult to define in practice due to the number of metallurgi-cal, chemical and mechanical variables that may affectthem, sound structural integrity assessment proceduresmust include proper stress analysis techniques for calculat-ing maximum tolerable flaw sizes. Such techniques areimportant in the design stage, but they are even moreuseful to evaluate operating structural components notoriginally designed for SCC service, when by any reasonthey must begin to work under such conditions dueto some unavoidable operational change (e.g. a pipe-line that must transport originally unforeseen amountsof H2S due to changes in oil well conditions while anew one specifically designed for such service is builtand commissioned.) Economical pressures to take sucha structural risk may be inescapable, because loss ofprofits associated with the very long time requiredfor replacing the component can be huge, especiallyin offshore applications. Such risky decisions can inprinciple be controlled by the methodology proposedas follows, which extends to EAC problems the analysisdeveloped to mechanically quantify notch sensitivityeffects through the behaviour of short fatigue cracks.



Indeed, if cracks behave well under SCC conditions, thatis, if fracture mechanics concepts can be used to describethem, then a ‘short crack characteristic size under SCCconditions’ can be defined by59

a0SCC ¼ 1=πð Þ� KISCC= η�SSCCð Þ½ �2: (30)

In this way, if all chemical effects involved in SCCproblems can be as usual assumed to be properly de-scribed and quantified by the traditional material resis-tances to crack initiation and propagation in the servicemedium under fixed stress conditions, SSCC and KISCC,supposing such pairs remain fixed, the a0 concept inSCC follows exactly the same idea of its analogous shortcrack characteristic size so useful for fatigue purposes: itmatches the otherwise separated material resistancesKISCC and SSCC to describe the behaviour of mechanicallyshort cracks and as so can potentially be equally useful inSCC problems. Such resistances are well-defined mate-rial properties for a given environment and can be mea-sured by standard procedures. Moreover, although SCCproblems are time dependent, SSCC and KISCC are not,because they quantify the limit stresses required forstarting or for growing cracks under SCC conditions.Hence, supposing that the mechanical parameters thatlimit SCC damage behave analogously to the equivalentparameters ΔKthR and ΔSLR that limit fatigue damage, aKitagawa-like diagram can be used to quantify the cracksizes a tolerable by any given component that works inSCC conditions under a given tensile stress σ, see Fig. 15.

This idea makes sense as well if KISCC and SSCC areviewed as the limits as R→ 1 for ΔKthR and ΔSLR in thegiven medium and can be further expanded. For example,Fig. 16 presents an extended Kitagawa–Takahashi dia-gram that shows four regions that may contain non-prop-agating cracks. First, starting from the bottom, the regionbounded by the material resistances to crack initiationand large crack growth by fatigue in a given aggressive

medium ΔSLR and ΔKthR/√(πa), which limits the toler-ance zone that may contain non-propagating fatiguecracks in that environment under fixed range loads at agiven R ratio; second, the region limited by SSCC andKISCC/√(πa) that may contain non-propagating cracksby SCC in that medium; third, the crack toleranceregion limited by ΔSLvac and ΔKthvac, the R-independentfatigue limit and FCG threshold of the given material invacuum; and fourth, the region limited by the intrinsicmaterial properties SUvac and KICvac/√(πa), which canonly be measured in vacuum or in truly inert environ-ments. The main advantage of looking at this problemin such an integrated way is that it turns natural theattempt to treat mechanical and chemical damage undera unified analysis procedure, following, for example,Vasudevan and Sadananda’s Unified Approachmethodologies.60,61

In other words, if cracks loaded under SCC conditionsbehave as they should, that is, if their mechanical drivingforce is indeed the SIF applied on them; and if the chem-ical effects that influence their behaviour are completelydescribed by the material resistance to crack initiationfrom smooth surfaces quantified by SSCC and by itsresistance to crack propagation measured by KISCC; then,it can be expected that cracks induced by SCC maydepart from sharp notches and then stop, due to thestress gradient ahead of the notch tips, eventuallybecoming non-propagating cracks, exactly as in thefatigue case. In such cases, the size of non-propagatingshort cracks can be calculated using the same proceduresused for fatigue, and the tolerance to such defects can beproperly quantified using an SCC notch sensitivity factorin structural integrity assessments. Hence, a criterion forthe maximum tolerable stress under SCC conditions canbe proposed as follows:

Fig. 16 Extended Kitagawa diagram including fatigue and stress cor-rosion cracking limiting conditions for crack growth.

Fig. 15 A Kitagawa–Takahashi-like diagram proposed to describethe environmentally assisted cracking behaviour of short and deepflaws for structural design purposes.



σmax≤KISCC=ffiffiffiffiffiπa

p �g a=wð Þ� 1þ a0SCC=að Þγ=2h i1=γ� �

;

a0SCC¼ 1=πð Þ KISCC= η�SSCCð Þ½ �2: (31)

In the same way, an expression analogous to Eq. (25) canbe used to properly define a ‘notch sensitivity under EACconditions’ by solving for a given γ (if it is necessary to betterfit the data) the system {φ/g=1, ∂(φ/g)∂x=0} for severalnotch tip radii ρ using κ≡KISCC/(SSCC√ρ) to obtain

qSCC κ; γð Þ≡ KtSCC κ; γð Þ � 1½ �= Kt � 1ð Þ (32)

where qSCC and KtSCC=1+ qSCC(Kt� 1) are the notchsensitivity and the effective SCF under EAC conditions. Inthis way, qSCC and KtSCC can be seen as analogous to the qand Kf parameters used for stress analyses under fatigueconditions.

Such equations can be used for stress analyses ofnotched components under SCC conditions. Hence, theyare potentially useful for structural design purposes whenoverconservative pass/non-pass criteria used to ‘solve’most practical SCC problems nowadays are not afford-able or cannot be used for any other reason. In fact, theycan form the basis for a mechanical criterion for SCC thatcan be applied even by structural engineers, because itdoes not require much expertise in chemistry to be use-ful. Moreover, they can be properly tested, as follows.

First, following expert advice (Vasudevan, privatecommunication), the basic SCC resistances were mea-sured for the Al 2024 – liquid gallium pair (Ga is liquidabove 30 °C, but curiously it only boils at 2204 °C). Themain advantage of this exotic material-environment pairis its very quick SCC (in fact, LME) reactions, in the or-der of minutes. In comparison, SCC-sensitive Al alloysmay take weeks to crack in NaCl-water solutions. More-over, contrary to other liquid metals that may cause LMEsuch as mercury, Ga is a safe, non-toxic material.

This 2024 Al alloy was originally obtained in a T351temper as a 12.7mm thick plate, with analysed composi-tion Al plus 4.44Cu, 1.35Mg, 0.54Mn, 0.18Zn, 0.16Fe,0.12Si, 0.02Cr, 0.01Zr and less than 0.05 of other ele-ments in weight percentage. However, the alloy had tobe annealed to remove its residual stresses, because inthe original as-received plate condition, the Ga environ-ment would induce the test specimens to break duringmanipulation. All test specimens were cut on theTransversal-Longitudinal (TL) direction of the plate,identified by metallographic procedures. The basicmechanical properties of the annealed 2024 Al alloy weremeasured following ASTM E8M standard procedures at35 °C, resulting in E = 70GPa, SY = 113MPa, SU = 240MPa and ultimate strain εU = 16%.

Stress corrosion cracking sensibility and reaction rates ofthe Al–Ga pair were qualitatively evaluated also at 35 °C in

very slow dε/dt = 4.5× 10�8/s strain rate tension tests madein servo-controlled electromechanical testing machines,following ASTM G129 and the NACE International, theCorrosion Society recommendations. The liquefied Gawas applied on the test specimens surfaces with a brush,and light bulbs were used to maintain the warm 35 °C tem-perature during the tests (Fig. 17). To guarantee that theexposure time was long enough to ensure the LMEphenomenon, the time necessary to propagate a crack inthe annealed Al 2024 – liquid Ga pair was double-checkedby testing precracked C(T) specimens such as those usedto measure KISCC.

Two such specimens were tested under 7.5MPa√m,and two others under 12MPa√m. The latter failed inless than 3 h, whereas the others did not fail after 2 days.So, following standard procedures and assuming that theincubation time should be a value close to 3 h, a preloadof 7.5MPa√m was applied for 1 day on the test speci-mens used for measuring KISCC. Similarly, a preload of30MPa was applied for 1 day on the test specimens usedto measure SSCC. Such basic SCC resistances weremeasured using incremental load steps induced bycalibrated load rings following ASTM E1681, ASTMF1624 and ISO 7539 standard procedures: SSCC testsstarted at 30MPa and used 2.5MPa steps and KISCC testsinitiated at 7.5MPa√m and used 0.25MPa√m steps.

Fig. 17 Load ring and warming light bulbs used for the stress corro-sion cracking tests.

Fig. 18 Initially smooth test specimen with 6.35mm diameter, usedfor measuring SSCC, the resistance to crack initiation under SCC,according to ASTM F1624 and ISO 7539 standards.



The time between successive load steps was at least onehour. The measured values were SSCC = 43.6 ± 4.2MPa(average of nine samples, with 95% reliability) andKISCC = 8.8 ± 0.3MPa√m (eight samples, 95% reliability).Some of the test specimens broken during such standardSCC tests are shown in Figs 18 and 19.

Finally, using such standard SCC properties, fourpairs of C(T)-like notched test specimens were designedto support a maximum local stress σ≅ 90MPa> 2�SSCCat their notch tips. The dimensions chosen for suchnotches were {b, ρ, b/w} = {20mm, 0.5mm, 0.33},{12mm, 0.5mm, 0.2}, {20mm, 0.2mm, 0.33}, {40mm,4.5mm, 0.67}, respectively for specimens TS1–TS2,TS3–TS4, TS5–TS6 and TS7–TS8, where b and ρ arethe notch depth and tip radius, and w is the specimenwidth, with both b and w measured from the load line.The idea was, of course, to study their SCF/stress gradi-ent combinations in order to assure tolerance to the shortcracks that should start at the tips of their notches, be-cause they were loaded well above SSCC. The (different)loads applied on each one of such notched test specimenswere maintained for at least 48 h.

Despite being submitted to a much longer exposurethan that required to measure SSCC and KISCC

according to standard procedures, none of suchnotched specimens failed during the tests, exactly aspredicted beforehand. A pair of notched test speci-mens is shown in Fig. 20. Figure 21 shows some ofthe unbroken notched test specimens after beingloaded under a maximum local stress at the notch tiphigher than twice the material resistance to crackinitiation under SCC conditions, σmax> 2�SSCC, for aperiod 50 times longer than the one from the SSCCmeasurement tests.

Fig. 19 C(T) specimen with w = 60 mm used for measuring KISCC,

the resistance to crack propagation under SCC, according to ASTME1681, ASTM F1624, and ISO 7539 procedures.

(a) (b)

Fig. 20 (a) One of the four pairs of notched C(T)-like specimens used for testing the stress corrosion cracking notch sensitivity predictions; and(b) metrological verification of the notch dimensions.



CONCLUS IONS

A generalized ETS parameter was used to model the de-pendence of the threshold stress intensity range for shortfatigue cracks on the crack size, as well as the behaviourof non-propagating cracks induced by EAC. This depen-dence was used to estimate the notch sensitivity factor qof shallow and of elongated notches both for fatigueand for EAC conditions, from studying the propagationbehavior of short non-propagating cracks that might ini-tiate from their tips. It was found that the notch

sensitivity of elongated slits has a very strong depen-dence on the notch aspect ratio, defined by the ratioc/b of the semi-elliptical notch that approximates theslit shape having the same tip radius. These predic-tions were calculated by numerical routines and veri-fied by proper experiments. On the basis of thispromising performance, a criterion to evaluate the in-fluence of small or large surface flaws in fatigue and inEAC problems was proposed. Such results indicatethat notch sensitivity can indeed be properly treatedas a mechanical problem.

Fig. 21 Notched C(T)-like specimens, all with the same width w = 60mm and, from top to bottom, with {b, r, b/w} = {20mm, 0.5mm, 0.33},{12mm, 0.5mm, 0.2}, {20mm, 0.2mm, 0.33}, and {40mm, 4.5mm, 0.67}, after being tested under Smax = 90MPa > 2×SSCC at the notch tip, ortwice the stress that would lead unnotched specimens to fail by SCC.



Acknowledgements

CNPq, the Brazilian Research Council, and Petrobras, theBrazilian oil company, have provided research scholarshipsfor some of the authors; Dr A. Vasudevan from ONR, theOffice of Naval Research of the US Navy, has contrib-uted with many stimulating discussions; and ONR hasprovided a grant to partially support this research.

REFERENCES

1 McEvily, A. J. (1988) The growth of short fatigue cracks: areview. Mater. Sci. Res. Int., 4, 3–11.

2 Sadananda, K. and Vasudevan, A. K. (1997) Short crack growthand internal stresses. Int. J. Fatig., 19, S99–S108.

3 Lawson, L., Chen, E. Y. and Meshii, M. (1999) Near-thresholdfatigue: a review. Int. J. Fatig., 21, S15–S34.

4 Navarro, A. and de los Rios, E. R. (1988) A microstructurally-short fatigue crack growth equation. Fatig. Fract. Eng. Mater.Struct., 11, 383–396.

5 Chapetti,M.D. (2003) Fatigue propagation threshold of short cracksunder constant amplitude loading. Int. J. Fatig., 25, 1319–1326.

6 Krupp, U., Düber, O., Christ, H. J., Künkler, B., Schick, A.and Fritzen, C. P. (2004) Application of the EBSD tech-nique to describe the initiation and growth behaviour ofmicrostructurally short fatigue cracks in a duplex steel.J. Microsc., 213, 313–320.

7 Chapetti, M. D. (2008) Fatigue assessment using an integratedthreshold curve method – applications. Eng. Fract. Mech., 75,1854–1863.

8 Verreman, Y. (2008) Propagation des fissures curtes. In: Fatiguedes Matériaux et Structures, vol. 2 (Edited by C. Bathias and A.Pineau), Hermes-Lavoisier, Cachan, France.

9 Lorenzino, P. and Navarro, A. (2013) Initiation and growthbehaviour of very-long microstucturally short fatigue cracks.Frattura ed Integrità Strutturale, 25, 138–144.

10 El Haddad, M. H., Topper, T. H. and Smith, K. N. (1979)Prediction of non-propagating cracks.Eng. Fract.Mech.,11, 573–584.

11 El Haddad, M. H., Smith, K. N. and Topper, T. H. (1979) Fatiguecrack propagation of short cracks. J. Eng. Mater. Tech., 101, 42–46.

12 Yu, M. T., Duquesnay, D. L. and Topper, T. H. (1988) Notchfatigue behavior of 1045 steel. Int. J. Fatig., 10, 109–116.

13 Kitagawa, H. and Takahashi, S. (1976) Applicability of fracturemechanics to very small crack or cracks in the early stage. SecondInt Conf Mech Behavior of Mat, Boston, MA, 627–631, ASM.

14 Forman, R. G., Shivakumar, V., Mettu, S. R. and Newman, J. C.(2000) Fatigue crack growth computer program NASGROversion 3.0, Reference Manual, NASA.

15 Bazant, Z. P. (1997) Scaling of quasibrittle fracture: asymptoticanalysis. Int. J. Fract., 83, 19–40.

16 Tanaka, K., Nakai, Y. and Yamashita, M. (1981) Fatigue growththreshold of small cracks. Int. J. Fract., 17, 519–533.

17 Livieri, P. and Tovo, R. (2004) Fatigue limit evaluation ofnotches, small cracks and defects: an engineering approach.Fatig. Fract. Eng. Mater. Struct., 27, 1037–1049.

18 DuQuesnay,D. L., Yu,M.T. andTopper, T.H. (1988) An analysisof notch-size effects at the fatigue limit. J. Test. Eval., 16, 375–385.

19 Vallellano, C., Navarro, A. and Dominguez, J. (2000) Fatiguecrack growth threshold conditions at notches. Part I: theory.Fatig. Fract. Eng. Mater. Struct., 23, 113–121.

20 Atzori, B., Lazzarin, P. and Filippi, S. (2001) Cracks andnotches: analogies and differences of the relevant stress distribu-tions and practical consequences in fatigue limit predictions. Int.J. Fatig., 23, 355–362.

21 Atzori, B., Lazzarin, P. and Meneghetti, G. (2003) Fracturemechanics and notch sensitivity. Fatig. Fract. Eng. Mater. Struct.,26, 257–267.

22 Ciavarella, M. and Meneghetti, G. (2004) On fatigue limit in thepresence of notches: classical vs. recent unified formulations. Int.J. Fatig., 26, 289–298.

23 Atzori, B., Meneghetti, G. and Susmel, L. (2005) Materialfatigue properties for assessing mechanical components weak-ened by notches and defects. Fatig. Fract. Eng. Mater. Struct.,28, 83–97.

24 Atzori, B., Lazzarin, P. and Meneghetti, G. (2005) A unifiedtreatment of the mode I fatigue limit of components containingnotches or defects. Int. J. Fract., 133, 61–87.

25 Meggiolaro, M. A., Miranda, A. C. O. and Castro, J. T. P. (2007)Short crack threshold estimates to predict notch sensitivityfactors in fatigue. Int. J. Fatig., 29, 2022–2031.

26 Castro, J. T. P., Meggiolaro, M. A., Miranda, A. C. O., Wu, H.,Imad, A. and Nouredine, B. (2012) Prediction of fatigue crackinitiation lives at elongated notch roots using short crackconcepts. Int. J. Fatig., 42, 172–182.

27 Barsom, J. M. and Rolfe, S. T. (1999) Fracture and FatigueControl in Structures, 3rd edn. ASTM, West Conshohocken,PA, USA.

28 Newman, J. C. and Raju, I. (1984) Stress-intensity factor equa-tions for cracks in three-dimensional finite bodies subjected totension and bending loads, NASA TM-85793.

29 Góes, R. C. O., Castro, J. T. P. and Martha, L. F. (2013) 3Deffects around notch and crack tips. Int. J. Fatig., doi: 10.1016/j.ijfatigue.

30 Castro, J. T. P. and Leite, J. C. C. (2013) Does notch sensibilityexist in environmentally assisted cracking (EAC)? J. Mater. Res.Tech., 2, 288–295.

31 Schijve, J. (2001) Fatigue of Structures and Materials. Kluwer,Dordrecht, Netherlands.

32 Peterson, R. E. (1974) Stress Concentration Factors. Wiley,Hoboken, NJ, USA.

33 Frost, N. E., Marsh, K. J. and Pook, L. P. (1999) Metal Fatigue.Dover, Mineola, NY, USA.

34 Tada, H., Paris, P. C. and Irwin, G. R. (1985) The Stress Analysisof Cracks Handbook. Del Research, St. Louis, Mo, USA.

35 Miranda, A. C. O., Meggiolaro, M. A., Martha, L. F. and Castro,J. T. P. (2012) Stress intensity factor predictions: comparisonand round-off error. Comput. Mater. Sci., 53, 354–358.

36 Murakami, Y. (2002) Metal Fatigue: Effects of Small Defects andNon-Metallic Inclusions. Elsevier, Amsterdam, Netherlands.

37 Meggiolaro, M. A. and Castro, J. T. P. (2010) Automation of thefatigue design under variable amplitude loading using the ViDasoftware. Int. J. Struct. Integrity, 1, 94–103.

38 Taylor, D. (1999) Geometrical effects in fatigue: a unifyingtheoretical model. Int. J. Fatig., 21, 413–420.

39 Taylor, D. (2007) The Theory of Critical Distances. Elsevier.40 Susmel, L. (2008) The theory of critical distances: a review of its

applications in fatigue. Eng. Fract. Mech., 75, 1706–1724.41 Susmel, L. and Taylor, D. (2010) The Theory of Critical

Distances as an alternative experimental strategy for the deter-mination of KIC and ΔKth. Eng. Fract. Mech., 77, 1492–1501.

42 Wu, H., Imad, A., Nouredine, B., Castro, J. T. P. andMeggiolaro, M. A. (2010) On the prediction of the residualfatigue life of cracked structures repaired by the stop-holemethod. Int. J. Fatig., 32, 670–677.



43 Rabbe, P., Lieurade, H. P. and Galtier, A. (2000) Essais defatigue, partie I. Techniques de l’Ingénieur, Traité M4170.www.techniques-ingenieur.fr. (accessed on 16/01/2013).

44 Borrego, L. P., Ferreira, J. M., Pinho da Cruz, J. M. and Costa,J. M. (2003) Evaluation of overload effects on fatigue crackgrowth and closure. Eng. Fract. Mech., 70, 1379–1397.

45 Creager, M. and Paris, P. C. (1967) Elastic field equations forblunt cracks with reference to stress corrosion cracking. Int. J.Fract. Mech., 3, 247–252.

46 Inglis, C. E. (1913) Stress in a plate due to the presence of cracksand sharp corners. Phil. Trans. Roy. Soc. A, 215, 119–233.

47 McEvily, A. J. andWei, R. P. (1972) Corrosion Fatigue – Chemistry,Mechanics & Microstructure, vol. 2. NACE, Huston, TX, USA, pp.381–395.

48 Fontana, M. G. (1986) Corrosion Engineering. McGraw Hill,New York, NY, USA.

49 Corrosion, ASM Handbook, ASM, Materials Park, OH, USA,vol. 13. (1992).

50 Leis, B. N. and Eiber, R. J. (1997) Stress-corrosion cracking ongas-transmission pipelines: history, causes, and mitigation. FirstInt Business Conf Onshore Pipelines, Berlin, Germany.

51 Dietzel, W. (2001) Fracture mechanics approach to stress corro-sion cracking. Anales de Mecánica de la Fractura, 18, 1–7.

52 Lynch, S. P. (2003) Failures of engineering components due toenvironmentally assisted cracking. Pract. Fail. Anal., 3, 33–42.

53 Corrosion: Fundamentals, Testing, and Protection, ASM Handbook,ASM, Materials Park, OH, USA, vol. 13A. doi: 10.1016/j.ijfatigue.2013.11.016 (2003).

54 Lisagor, W. B. (2005) Environmental cracking – stress corro-sion, ASTM Corrosion Tests and Standards – Interpretationand Application (Manual 20).

55 Vasudevan, A. K. and Sadananda, K. (2009) Classification ofenvironmentally assisted fatigue crack growth behavior. Int. J.Fatig., 31, 1696–1708.

56 Sadananda, K. and Vasudevan, A. K. (2011) Review of environ-mentally assisted cracking.Metall. Mater. Trans. A, 42, 279–295.

57 Vasudevan, A. K. and Sadananda, K. (2011) Role of internalstresses on the incubation times during stress corrosion crack-ing. Metall. Mater. Trans. A, 42, 396–404.

58 Vasudevan, A. K. and Sadananda, K. (2011) Role of slip mode onstress corrosion cracking behavior. Metall. Mater. Trans. A, 42,405–414.

59 Castro, J. T. P. andMeggiolaro, M. A. (2013) Is notch sensitivitya stress analysis problem? Frattura ed Integrità Strutturale, 25,79–86.

60 Sadananda, K. and Vasudevan, A. K. (2011) Failure diagram forchemically assisted crack. Metall. Mater. Trans. A, 42, 296–303.

61 Sadananda, K. (2013) Failure diagram and chemical drivingforces for subcritical crack growth. Metall. Mater. Trans. A, 44,1190–1199.



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